Event-triggered fault detection filter design for nonlinear networked systems via fuzzy Lyapunov functions

Abstract

This paper investigates the problem of event-triggered fault detection filter design for nonlinear networked control systems with both sensor faults and process faults. First, Takagi–Sugeno (T–S) fuzzy model is utilized to represent the nonlinear systems with faults and disturbances. Second, a discrete event-triggered communication scheme is proposed to reduce the utilization of limited network bandwidth between filter and original system. At the same time, considering network-induced delays and event-triggered scheme, a novel T–S fuzzy fault detection filter is constructed to generate a residual signal, which has nonsynchronous premise variables with the original T–S fuzzy system. Then, the fuzzy Lyapunov functional based approach and the reciprocally convex approach are developed such that the obtained sufficient conditions ensure that the fuzzy fault detection system is asymptotically stable with H_∞ performance and is less conservative. All the conditions are given in terms of linear matrix inequalities (LMIs), which can be solved by LMI tools in MATLAB environment. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed results.

Introduction

With the increasing requirement for safety, reliability and higher performance of the industrial systems, the fault detection problems have become an important research area, and some outstanding achievements have been obtained [1], [2], [3]. For linear systems, the robust fault detection filter has been designed in [4], and an on-line detection method has been proposed for linear discrete-time uncertain systems [5], [6]. Considering the nonlinear characteristics of industrial systems, it is significant and difficult to study the fault detection problems of no nlinear systems. T–S fuzzy model approach, as a very flexible modeling approach for nonlinear systems, has been drawing a great deal of attention. Correspondingly, many results were achieved about the fuzzy control and fault detection, see [7], [8], [9], [10], [11], [12], where Li et al. [11] studied T–S fuzzy systems with sensor failures and a real-time weighted observer-based fault detection approach was proposed; Zhang et al. [12] extended the robust fault detection approach to switch fuzzy systems. In general, the common quadratic Lyapunov function was applied to the fuzzy system stability analysis and filter/controller design. The basis idea of it lies in lumping the design schemes into finding a common Lyapunov functions of a set of linear matrix inequalities for every subsystems, by which those stability results may be rather conservative and limit its application due to the difficulties in finding a common Lyapunov matrix [13]. To reduce the conservatism and improve the application, some effective analysis approaches were developed, such as, the piecewise Lyapunov functions [14], [15], [16] and fuzzy Lyapunov functions [17], [18], [19], [20], [21]. The fuzzy Lyapunov function is constructed from a set of positive definite matrices for each linear time-invariant submodel and the membership functions for the T–S fuzzy plant. Nevertheless, the presence of the time derivative of fuzzy weighting functions makes it more complicated to develop a suitable relaxation method for stability analysis and controller synthesis. Furthermore, some approaches were proposed in [17], [18] to reduce the conservatism progressively by introducing more slack matrices and exploring the properties of the fuzzy weighting functions and their time derivatives.

On the other hand, networked control systems (NCSs) have gained increasing attention because of their advantages, such as the low costs in maintenance and installation, higher reliability, and flexibility in communication architectures [22], [23], [24] and the references therein. In typical networks with limited bandwidth, simultaneous multiple transmissions over the networked system would give rise to data collisions [25]. Some effective ways for preventing that phenomenon are to orchestrate the transmission order of signal by the communication protocols and some significant results have been presented [26], [27], [28]. The stochastic protocol [27], round-robin and weighted try-once-discard protocols [28] were applied to the filtering problems of a class of time-varying nonlinear delayed systems. Network-induced delays and packet dropouts occur inevitably due to the shared communication network between sensors, controllers, and actuators. Many important results about fault detection for NCSs were achieved [29], [30], [31], [32]. In [29], the fault detection problem with signal quantization and random packet dropouts was investigated. In [30], a fault detection scheme for NCSs with finite-frequency servo inputs was proposed. In [31], a mode-dependent fault detection filter for NCSs was designed, in which the access status of sensors was modeled by Markovian jumping theory and transmission delays were considered simultaneously. In [32], a robust fault detection problem for a class of uncertain discrete-time T–S fuzzy systems with stochastic mixed time delays and successive packet dropouts was investigated. It should be mentioned that the network-induced delays and packet dropouts need to be considered in NCSs, which may lead to considerable analytical difficulties but be practically valuable for fault detection.

Note that in aforementioned works, the time triggered approach was considered such that all the sampled data packets were released into the communication network. However, some sampled data and measurement outputs in NCSs are not essential to be released, which is a waste of the limited network resource. The event-triggered scheme for NCSs is a alternative approach to utilize the limited communication bandwidth effectively [33], [34], [35], [36], [37], [38]. That is, the sampled-data is transmitted only when the predefined threshold is satisfied. In [36], a novel discrete event-triggered fuzzy filter was designed for a class of nonlinear NCSs. It is worth mentioning that the ellipsoidal triggering condition was proposed in [37], which is quite general that covers several well-studied triggering conditions as special cases. In [38], a novel ellipsoidal event-triggered scheme was applied to research consensus control problem for discrete time-varying stochastic multi-agent system. The event-triggered fault detection problems have also received considerable attention [39], [40], [41], [42], [43], [44]. The problem of event-triggered fault detection for discrete-time linear system was considered, and a Luenberger observer was designed to generate residual signals in [41]. Fuzzy polynomial event-triggered fault detection approach was proposed in [42]. The process faults and nonlinear perturbation were considered simultaneously in [43]. The event-triggered fault detection filter and controller were co-designed to detect sensor faults in [44]. It is pointed out that usually either process faults or sensor ones were considered for the fault detection problems under the event-triggered scheme for NCSs, little attention has been concentrated on both faults and disturbances simultaneously, which are inevitable in many practical plants. When one kind of disturbances or faults is ignored, the higher false alarm rate may appear and fault detection performance may worsen, which motivates this work.

This paper studies the problem of event-triggered fault detection filter design for nonlinear networked control systems with process faults, sensor faults and disturbances. The T–S fuzzy model is used for the nonlinear system. A discrete event-triggered scheme is proposed to reduce the utilization of limited network bandwidth. Then, the fuzzy fault detection filter with nonsynchronized premise variables with fuzzy model is designed under the cases of network-induced delays and packet dropouts.

The main contributions of this paper are summarized as follows.

• The fault detection problem simultaneously including process faults, sensor faults and disturbances is considered for nonlinear networked control systems based on the T–S fuzzy model approach.

• A novel discrete event-triggered fuzzy H_∞ fault detection filter is constructed, which is subject to network-induce transmission delays and nonsynchronized premise variables with original system.

• Fuzzy Lyapunov functions and reciprocally convex approach are developed such that some stability criteria in LMI forms are obtained, which ensure the fault detection system is asymptotically stable with H_∞ performance and is less conservative.

The remainder of this paper is organized as follows. In Section 2, the fault detection systems for NCSs based on event-triggered scheme is constructed. In Section 3, the fuzzy fault detection filter is designed with H_∞ performance. A numerical example is given in Section 4. Conclusion is presented in Section 5.

Notation: The superscripts “T” and “$-1$” stand for the transpose of matrix and the inverse of matrix, respectively. The notation P > 0 means that the matrix P is a real symmetric positive definite matrix and diag{⋅⋅⋅} denotes a block-diagonal matrix. The space of square integrable vector functions is denoted by L₂[0, ∞), and for ω(t) ∈ L₂[0, ∞) is given by $\parallel \omega \left(t\right){\parallel }_2=\sqrt{{\int }_0^{\infty }\mid \omega \left(t\right){\mid }^2}$. The col{ · } denotes the column vector of  ·  and * denote the matrix entries implied by symmetry.